A transference principle for involution-invariant functional Hilbert spaces

Abstract

Let $\ddot{I}ƒ:{\mathbb{C}}^d\to {\mathbb{C}}^d$ be an affine-linear involution such that $J_{\ddot{I}}ƒ=-1$ and let $U,V$ be two domains in ${\mathbb{C}}^d.$ Let $\ddot{I}•:U\to V$ be a $\ddot{I}ƒ$-invariant $2$-proper map such that $J_{\ddot{I}}•$ is affine-linear and let $\mathcal{H}(U)$ be a $\ddot{I}ƒ$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space ${\mathcal{H}}_{\ddot{I}}•(V):=\{f\in \mathrm{Hol}(V):J_{\ddot{I}}•\cdot f\circ \ddot{I}•\in \mathcal{H}(U)\}$ endowed with the norm $\Vert f{\Vert }_{\ddot{I}}•:=\Vert J_{\ddot{I}}•\cdot f\circ \ddot{I}•{\Vert }_{\mathcal{H}(U)}$ is a reproducing kernel Hilbert space and the linear mapping ${\Gamma }_{\ddot{I}}•$ defined by ${\Gamma }_{\ddot{I}}•(f)=J_{\ddot{I}}•\cdot f\circ \ddot{I}•,$ $f\in \mathrm{Hol}(V),$ is a unitary from ${\mathcal{H}}_{\ddot{I}}•(V)$ onto $\{f\in \mathcal{H}(U):f=-f\circ \ddot{I}ƒ\}.$ Moreover, a neat formula for the reproducing kernel $\hat{I}{º}_{\ddot{I}}•$ of ${\mathcal{H}}_{\ddot{I}}•(V)$ in terms of the reproducing kernel of $\mathcal{H}(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann's inequality for contractive tuples naturally associated with these domains.

Major revision